solving absolute value equations and inequalities

1402 Pages·2013·23.42 MB·English

Preface
Intermediate Algebra is the second part of a two-part course in Algebra. Written in a clear and
concise manner, it carefully builds on the basics learned in Elementary Algebra and introduces the
more advanced topics required for further study of applications found in most disciplines. Used as a
standalone textbook, it offers plenty of review as well as something new to engage the student in each
chapter. Written as a blend of the traditional and graphical approaches to the subject, this textbook
introduces functions early and stresses the geometry behind the algebra. While CAS independent, a
standard scientific calculator will be required and further research using technology is encouraged.
Intermediate Algebra clearly lays out the steps required to build the skills needed to solve a variety of
equations and interpret the results. With robust and diverse exercise sets, students have the
opportunity to solve plenty of practice problems. In addition to embedded video examples and other
online learning resources, the importance of practice with pencil and paper is stressed. This text
respects the traditional approaches to algebra pedagogy while enhancing it with the technology
available today. In addition, Intermediate Algebra was written from the ground up in an open and
modular format, allowing the instructor to modify it and leverage their individual expertise as a
means to maximize the student experience and success.
The importance of Algebra cannot be overstated; it is the basis for all mathematical modeling used in
all disciplines. After completing a course sequence based on Elementary and Intermediate Algebra,
students will be on firm footing for success in higher-level studies at the college level.
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
1 of 1402
Table of Contents
About the Author
Acknowledgments
Preface
Chapter 1: Algebra Fundamentals
1.1: Review of Real Numbers and Absolute Value
1.2: Operations with Real Numbers
1.3: Square and Cube Roots of Real Numbers
1.4: Algebraic Expressions and Formulas
1.5: Rules of Exponents and Scientific Notation
1.6: Polynomials and Their Operations
1.7: Solving Linear Equations
1.8: Solving Linear Inequalities with One Variable
1.9: Review Exercises and Sample Exam
Chapter 2: Graphing Functions and Inequalities
2.1: Relations, Graphs, and Functions
2.2: Linear Functions and Their Graphs
2.3: Modeling Linear Functions
2.4: Graphing the Basic Functions
2.5: Using Transformations to Graph Functions
2.6: Solving Absolute Value Equations and Inequalities
2.7: Solving Inequalities with Two Variables
2.8: Review Exercises and Sample Exam
Chapter 3: Solving Linear Systems
3.1: Linear Systems with Two Variables and Their Solutions
3.2: Solving Linear Systems with Two Variables
3.3: Applications of Linear Systems with Two Variables
3.4: Solving Linear Systems with Three Variables
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
2 of 1402
3.5: Matrices and Gaussian Elimination
3.6: Determinants and Cramer’s Rule
3.7: Solving Systems of Inequalities with Two Variables
3.8: Review Exercises and Sample Exam
Chapter 4: Polynomial and Rational Functions
4.1: Algebra of Functions
4.2: Factoring Polynomials
4.3: Factoring Trinomials
4.4: Solve Polynomial Equations by Factoring
4.5: Rational Functions: Multiplication and Division
4.6: Rational Functions: Addition and Subtraction
4.7: Solving Rational Equations
4.8: Applications and Variation
4.9: Review Exercises and Sample Exam
Chapter 5: Radical Functions and Equations
5.1: Roots and Radicals
5.2: Simplifying Radical Expressions
5.3: Adding and Subtracting Radical Expressions
5.4: Multiplying and Dividing Radical Expressions
5.5: Rational Exponents
5.6: Solving Radical Equations
5.7: Complex Numbers and Their Operations
5.8: Review Exercises and Sample Exam
Chapter 6: Solving Equations and Inequalities
6.1: Extracting Square Roots and Completing the Square
6.2: Quadratic Formula
6.3: Solving Equations Quadratic in Form
6.4: Quadratic Functions and Their Graphs
6.5: Solving Quadratic Inequalities
6.6: Solving Polynomial and Rational Inequalities
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
3 of 1402
6.7: Review Exercises and Sample Exam
Chapter 7: Exponential and Logarithmic Functions
7.1: Composition and Inverse Functions
7.2: Exponential Functions and Their Graphs
7.3: Logarithmic Functions and Their Graphs
7.4: Properties of the Logarithm
7.5: Solving Exponential and Logarithmic Equations
7.6: Applications
7.7: Review Exercises and Sample Exam
Chapter 8: Conic Sections
8.1: Distance, Midpoint, and the Parabola
8.2: Circles
8.3: Ellipses
8.4: Hyperbolas
8.5: Solving Nonlinear Systems
8.6: Review Exercises and Sample Exam
Chapter 9: Sequences, Series, and the Binomial Theorem
9.1: Introduction to Sequences and Series
9.2: Arithmetic Sequences and Series
9.3: Geometric Sequences and Series
9.4: Binomial Theorem
9.5: Review Exercises and Sample Exam
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
4 of 1402
About the Author
John Redden
John Redden earned his degrees at California State University–Northridge and Glendale Community
College. He is now a professor of mathematics at the College of the Sequoias, located in Visalia, California.
With over a decade of experience working with students to develop their algebra skills, he knows just
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
5 of 1402
where they struggle and how to present complex techniques in more understandable ways. His student-
friendly and commonsense approach carries over to his writing of Intermediate Algebra and various
other open-source learning resources.
Author site: http://edunettech.blogspot.com/
Acknowledgments
I would like to thank the following reviewers whose feedback helped improve the final product:
(cid:0) Katherine Adams, Eastern Michigan University
(cid:0) Sheri Berger, Los Angeles Valley College
(cid:0) Seung Choi, Northern Virginia Community College
(cid:0) Stephen DeLong, Colorado Mountain College
(cid:0) Keith Eddy, College of the Sequoias
(cid:0) Solomon Emeghara, William Patterson University
(cid:0) Audrey Gillant, SUNY–Maritime
(cid:0) Barbara Goldner, North Seattle Community College
(cid:0) Joseph Grich, William Patterson University
(cid:0) Caroll Hobbs, Pensacola State College
(cid:0) Clark Ingham, Mott Community College
(cid:0) Valerie LaVoice, NHTI, Concord Community College
(cid:0) Sandra Martin, Brevard Schools
(cid:0) Bethany Mueller, Pensacola State College
(cid:0) Tracy Redden, College of the Sequoias
(cid:0) James Riley, Northern Arizona University
(cid:0) Bamdad Samii, California State University–Northridge
(cid:0) Michael Scott, California State University–Monterey Bay
(cid:0) Nora Wheeler, Santa Rosa Junior College
I would also like to acknowledge Michael Boezi and Vanessa Gennarelli of Flat World Knowledge.
The success of this project is in large part due to their vision and expertise. Finally, a special heartfelt
thank-you is due to my wife, Tracy, who spent countless hours proofreading and editing these pages
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
6 of 1402
—all this while maintaining a tight schedule for our family. Without her, this textbook would not
have been possible.
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
7 of 1402
Chapter 1
Algebra Fundamentals
Review of Real Numbers and Absolute Value
1.1
LEARNING OBJECTIVES
1. Review the set of real numbers.
2. Review the real number line and notation.
3. Define the geometric and algebraic definition of absolute value.
Real Numbers
Algebra is often described as the generalization of arithmetic. The systematic use
of variables, letters used to represent numbers, allows us to communicate and solve a
wide variety of real-world problems. For this reason, we begin by reviewing real
numbers and their operations.
A set is a collection of objects, typically grouped within braces { }, where each object is
called an element. When studying mathematics, we focus on special sets of numbers.
The three periods (…) are called an ellipsis and indicate that the numbers continue
without bound. A subset, denoted , is a set consisting of elements that belong to a given
set. Notice that the sets of natural and whole numbers are both subsets of the set of
integers and we can write:
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
8 of 1402
A set with no elements is called the empty set and has its own special notation:
Rational numbers , denoted  Q, are defined as any number of the form where a and b are
integers and b is nonzero. We can describe this set using set notation :
The vertical line | inside the braces reads, “such that” and the symbol ∈indicates set membership
and reads, “is an element of.” The notation above in its entirety reads, “the set of all
numbers such that a and b are elements of the set of integers and b is not equal to zero.”
Decimals that terminate or repeat are rational. For example,
he set of integers is a subset of the set of rational numbers, Z⊆Q, because every integer can be
expressed as a ratio of the integer and 1. In other words, any integer can be written over 1 and can be
considered a rational number. For example,
Irrational numbers are defined as any numbers that cannot be written as a ratio of two integers.
Non terminating decimals that do not repeat are irrational. For example,
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
9 of 1402
Finally, the set of real numbers , denoted R, is defined as the set of all rational numbers combined
with the set of all irrational numbers. Therefore, all the numbers defined so far are subsets of the set
of real numbers. In summary,
The set of even integers is the set of all integers that are evenly divisible by 2. We can obtain the
set of even integers by multiplying each integer by 2.
The set of odd integers is the set of all nonzero integers that are not evenly divisible by 2.

A prime number is an integer greater than 1 that is divisible only by 1 and itself. The smallest
prime number is 2 and the rest are necessarily odd.

Any integer greater than 1 that is not prime is called a composite number and can be uniquely
written as a product of primes. When a composite number, such as 42, is written as a
product, 42=2⋅21, we say that 2⋅21 is a factorization of 42 and that 2 and 21 are factors. Note that
Attributed to John Redden
Saylor URL: http://www.saylor.org/books/
10 of 1402